We wish to construct a rectangular auditorium with a stage shaped as a semicircle of radius $r$, as shown in the diagram below (white is the stage and green is the seating area). For safety reasons, light strips must be placed on the perimeter of the seating area. If we have $45\pi + 60$ meters of light strips, what should $r$ be so that the seating area is maximized?
So I first set the width of the seating area to 2r, and the depth to be x. The perimeter would then be $2x + 2r + \pi r$ = $45\pi + 60$. The problem asks us to maximize the area, though, so it's $2rx – (\pi r^2)/2$. I can solve the equation in terms of x so that it becomes $x = (45\pi + 60 – r(\pi + 2))/2$. Unfortunately, I'm stuck from this point on, so any hint that you could give me would be great. Thanks!
Best Answer
The maximization problem you want to solve is $$ \frac{1}{2}\max_{2x+(2+\pi)r=45\pi+60} r(4x-\pi r) $$ or $$ \frac{1}{8+6\pi}\max_{4x+(4+2\pi)r=90\pi+120} (4+3\pi)r(4x-\pi r). $$ By letting $A=(4+3\pi)r$ and $B=4x-\pi r$ this can be written as $$ \frac{1}{8+6\pi}\max_{A+B=90\pi+120} AB $$ and by the AM-GM inequality it equals $$ \frac{1}{8+6\pi}\left(60+45\pi\right)^2 = \frac{225}{2}(4+3\pi),$$ achieved by $r=15$.